King Fahd University of Petroleum and Minerals
Department of Mathematics and Statistics
Workshop on
Fractional Models in Science & Engineering (FMSE18)
Theory and Computation

Monday, December 10, 2018
KFUPM Conference Room (Building # 20)
             
Abdul Khaliq
Department of Mathematical Sciences & Computational Science
Middle Tennessee State University, USA.
Email: Abdul.Khaliq@mtsu.edu

A split-step second order predictor-corrector method for space-fractional reaction-diffusion equations with nonhomogeneous boundary conditions is presented and Matrix Transfer Technique (MTT) is used for spatial discretization. The method is shown to be unconditionally stable and second order convergent. Numerical experiments are performed to confirm the stability and second order convergence of the method. The split-step predictor-corrector method is also compared with an IMEX predictor-corrector method which is found to incur oscillatory behavior for some time steps. Our method is seen to produce reliable and oscillation free results for any time step when implemented on numerical examples with no smooth initial data. We also present a priori reliability constraint for IMEX predictor-corrector method to avoid unwanted oscillations and show its validity numerically on several test problems from the literature.


Roberto Garrappa
Department of Mathematics
University of Bari, Italy.

Email: roberto.garrappa@uniba.it

On special fractional operators in computational electromagnetism: theory and numerical methods.

The propagation of electric and magnetic fields in complex systems, such as for instance special polymers or biologic tissues, is usually described in the frequency domains by means of functions with more than one real powers. The characterization of these models in the time domain involves new non-standard differential operators of fractional order [1]. In this talk we discuss the constitutive law of the Havriliak-Negami model which allows to close the Maxwell's system and describe the interactions between electric and magnetic fields in a wide range of materials, with important applications in engineering and medicine. After describing new recent developments for characterizing in the time domain the operators associated with this model, we focus with the connections with fractional integrals and fractional derivatives based on the so-called Prabhakar function [2] and we discuss some of the possible approaches for the discretization of these operators and for their use in numerical simulations [3].  

[1] R. Garrappa, F. Mainardi, G. Maione, Models of dielectric relaxation based on completely monotone
     functions.
  Fract. Calc. Appl.  Anal., 2016, 19(5), 1105-1160 
[2] R. Garra, R. Garrappa, The Prabhakar or three parameter Mittag--Leffler function: theory and application.
     Commun. Nonlinear Sci. Numer. Simul., 2018, 56, 314-329
 
[3] R. Garrappa, On Grunwald-Letnikov operators for fractional relaxation in Havriliak-Negami models, Commun.
     Nonlinear Sci. Numer.
  Simul., 2016, 38, 178-191

 

Tenreiro Machado 
Department of Electrical & Computer Engineering
University of Porto, Portugal.
Email:
jtm@isep.ipp.pt
 

Fractional Calculus: The Perspective of Complex Systems.

 

Fractional Calculus (FC) started in 1695 when L'HŰpital wrote a letter to Leibniz asking for the meaning of Dny for n = 1/2. Starting with the ideas of Leibniz many important mathematicians developed the theoretical concepts. By the beginning of the twentieth century Olivier Heaviside applied FC in electrical engineering, but his visionary and important contributions were forgotten during several decades. Only in the eighties FC emerged associated with phenomena such as fractal and chaos and, consequently, in nonlinear dynamical. In the last years, FC become a 'new' mechanism for the analysis of dynamical systems. FC is now recognized to be an important tool to model and control systems with long range memory effects. This lecture introduces several applications in distinct areas of science and engineering based on the authorís own experience and research work during the last years. The presentation focusses advanced topics, out of the standard stream, usually followed by the scientific community, where the FC and complex systems methods reveal important properties.

 

 

 

Salman Malik

Department of Mathematics
COMSATS University, Pakistan.

Email: salman.amin.malik@gmail.com

 

 

Inverse Problems for Some Space-Time Fractional Differential Equations.

 

Fractional Calculus (FC), that is, the study of arbitrary order integrals and derivatives has beenconsidered by many experts from all fields of sciences. FC has applications in many fields just tomention a few are in biology, physics, finance, viscoelasticity processes, prediction of extreme eventslike earthquake etc. The fractional derivatives in time and space have been considered to explain theanomalies in the complex phenomena of diffusion/transport at both micro and macro level. The timefractional diffusion equations with Riemann-Liouville or Caputo fractional derivatives are equivalentto infinitesimal generators of time fractional evolutions that arise in the transition from microscopicto macroscopic time scales. The transition from first order derivative to the fractional order timederivative arise physically reported by many see for example [1].

Inverse problems for fractional differential equations have been considered by many in the recentpast which include inverse source problems, inverse coefficient problems etc. Indeed, the inverse problemsfor time, space and time-space fractional diffusion equations are considered. In this talk weare going to discuss the inverse source problems for the Space-Time Fractional Differential Equations(STFDEs). The STFDE is obtained from classical diffusion equation by replacing time derivativewith fractional order time derivative and Sturm-Liouville operator by fractional order Sturm-Liouvilleoperator. The spectral problem of the STFDEs is the fractional order Sturm-Liouville system, whichis a generalization of the well known Sturm-Liouville system involving fractional derivatives in spacedefined in Caputoís sense with Dirichlet boundary conditions. By variational techniques several properties of eigenvalues and eigenfunctions of the fractional order Sturm-Liouville system were investigated in [2].

In the first part of my talk, an inverse problem of recovering a space dependent source term along with solution for a STFDE from over-specified condition of final data will be discussed [3].

The conditions on the given data such that a unique solution of the inverse problem exists will be presented. Secondly, inverse problem of determining a time dependent source term from the total energy measurement of the system (the over-specified condition) for a STFDE will be considered [4].

The existence and uniqueness results are proved by using eigenfunction expansion method. Several special cases, particular examples will be provided. Some future perspectives will be shared with the audience.

 

References
  1. R. Hilfer, Applications of Fractional Calculus in Physics, World Scienti c, 2000.
  2. M. Klimek, A.B. Malinowska, T. Odzijewicz, Variational methods for the fractional Sturm-Liouville
    problem, J. Math. Anal. Appl., 416, (2014), 402-428.
  3. M. Ali, S. Aziz and S.A. Malik, Inverse source problem for a space-time fractional diffusion equation,
    Fract. Calc. Appl. Anal., 21 (2018), 844-863.
  4. M. Ali, S. Aziz, S.A. Malik, Inverse problem for a space-time fractional diffusion equation: Application
    of fractional Sturm-Liouville operator, Mathematical Methods in the Applied Sciences, 41 (2018),
    2733-2744.

 

Ozgur Kirlangic
Saudi Aramco Expec ARC Computational Modeling Technology (CMT) Team

Email: ozgur.kirlangic@aramco.com

Fracture modeling in the current reservoir simulation applications and potential for fractional model

Numerical simulation of fractured reservoirs is a challenging task due to the huge difference between the scales of velocity in the fractures and the rock matrix. In fractured reservoirs, the matrix contributes most of the fluid storage but very little on fluid transport; on the contrary, the fracture system provides high flow capacity but low pore volume [1]. A common approach in dealing with this problem is applying dual or multi-porosity models, which is based on separation of scales. This talk will be about the current state of art in the applied fractured reservoir simulation and explore where the fractional diffusion equations would fit and make difference for the future applications.  [1] Bicheng Yan, Masoud Alfi, Cheng An, Yang Cao, Yuhe Wang, John E. Killough, General Multi-Porosity simulation for fractured reservoir modeling, Journal of Natural Gas Science and Engineering, Volume 33, 2016, Pages 777-791